What is the role of the 'fuzzifier' w in Fuzzy Clustering?
Its not required, you can have $m=1$, actually it can be any number $\geq 1$.   

Now the better question is why to have it? The answer is that it adds a smoothing effect. Lets look at it in each of the limits ($\lim m \rightarrow 1$ and $\lim m \rightarrow \infty$)  

Towards $\infty$, it makes $u_{ij}$ equal to $\frac{1}{c}$, making each point have equal membership of each class regardless of $m$. From the optimization perspective, its saying how can we achieve finding clusters that are closest to all points, therefore by definition it has already achieved that, and so the Loss will always be 0. (at its global minimum)  

Now in the other limit, the constants are inversely proportional to the square of the normalized euclidean distance. This makes intuitive sense, the membership is high if they are close, and the membership is low if they are not (relatively)  

So why do we have the $m$, its for control. It allows us to choose and experiment with how heavy each distance should hold weight in the membership. An example where a larger $m$ may be useful is when the data isnt clean, and you dont want to rely so heavily on euclidean distance as the membership, so you forcibly add in a smoothing effect